Sir Andrew J. Wiles Awarded Abel Prize Elaine Kehoe with The Norwegian Academy of Science and Letters official Press Release ©Abelprisen/DNVA/Calle Huth. Courtesy of the Abel Prize Photo Archive. ©Alain Goriely, University of Oxford. Courtesy the Abel Prize Photo Archive. Sir Andrew Wiles received the 2016 Abel Prize at the Oslo award ceremony on May 24. The Norwegian Academy of Science and Letters has carries a cash award of 6,000,000 Norwegian krone (ap- awarded the 2016 Abel Prize to Sir Andrew J. Wiles of the proximately US$700,000). University of Oxford “for his stunning proof of Fermat’s Citation Last Theorem by way of the modularity conjecture for Number theory, an old and beautiful branch of mathemat- semistable elliptic curves, opening a new era in number ics, is concerned with the study of arithmetic properties of theory.” The Abel Prize is awarded by the Norwegian Acad- the integers. In its modern form the subject is fundamen- tally connected to complex analysis, algebraic geometry, emy of Science and Letters. It recognizes contributions of and representation theory. Number theoretic results play extraordinary depth and influence to the mathematical an important role in our everyday lives through encryption sciences and has been awarded annually since 2003. It algorithms for communications, financial transactions, For permission to reprint this article, please contact: and digital security. [email protected]. Fermat’s Last Theorem, first formulated by Pierre de DOI: http://dx.doi.org/10.1090/noti1386 Fermat in the seventeenth century, is the assertion that 608 NOTICES OF THE AMS VOLUME 63, NUMBER 6 the equation xn+yn=zn has no solutions in positive integers tophe Breuil, Brian Conrad, Fred Diamond, and Richard for n>2. Fermat proved his claim for n=4, Leonhard Euler Taylor. As recently as 2015, Nuno Freitas, Bao V. Le Hung, found a proof for n=3, and Sophie Germain proved the and Samir Siksek proved the analogous modularity state- first general result that applies to infinitely many prime ment over real quadratic number fields. Few results have exponents. Ernst Kummer’s study of the problem unveiled as rich a mathematical history and as dramatic a proof as several basic notions in algebraic number theory, such as Fermat’s Last Theorem. ideal numbers and the subtleties of unique factorization. The complete proof found by Andrew Wiles relies on three Biographical Sketch further concepts in number theory, namely elliptic curves, Andrew J. Wiles was born April 11, 1953, in Cambridge, modular forms, and Galois representations. United Kingdom. He earned his PhD in 1980 at Clare Col- Elliptic curves are defined by cubic equations in two lege, Cambridge. He has held positions at Harvard Univer- variables. They are the natural domains of definition of sity and Princeton University. In 1985–86, he was a Guggen- the elliptic functions introduced by Niels Henrik Abel. heim Fellow at the Institut des Hautes Études Scientifiques Modular forms are highly symmetric analytic functions and at the École Normale Supérieure. From 1988 to 1990 he defined on the upper half of the complex plane, and was a Royal Society Research Professor at the University of naturally factor through shapes known as modular curves. Oxford before returning to Princeton. He rejoined Oxford An elliptic curve is said to be modular if it can be param- in 2011 as Royal Society Research Professor. etrized by a map from one of these modular curves. The Andrew Wiles is one of the very few mathematicians— modularity conjecture, proposed by Goro Shimura, Yutaka if not the only one—whose proof of a theorem has been Taniyama, and André Weil in the 1950s and 1960s, claims international headline news. His proof was not only the that every elliptic curve defined over the rational numbers high point of his career—and an epochal moment for is modular. mathematics—but also the culmination of a remark- In 1984, Gerhard Frey associated a semistable elliptic able personal journey that began three decades before. curve to any hypothetical counterexample In 1963, when he was a ten-year-old boy to Fermat’s Last Theorem, and strongly growing up in Cambridge, Wiles found a suspected that this elliptic curve would I knew...I would copy of a book on Fermat’s Last Theorem not be modular. Frey’s nonmodularity was in his local library. He became captivated proven via Jean-Pierre Serre’s epsilon con- never let it go. I by the problem—that there are no whole jecture by Kenneth Ribet in 1986. Hence, a number solutions to the equation xn+yn=zn proof of the Shimura-Taniyama-Weil mod- had to solve it. when n is greater than 2—which was easy ularity conjecture for semistable elliptic to understand but which had remained curves would also yield a proof of Fermat’s unsolved for three hundred years. “I knew Last Theorem. However, at the time the modularity conjec- from that moment that I would never let it go,” he said. ture was widely believed to be completely inaccessible. It “I had to solve it.” was therefore a stunning advance when Andrew Wiles, in Wiles studied mathematics at Merton Col- a breakthrough paper published in 1995, introduced his lege, Oxford, and returned to Cambridge, at Clare modularity lifting technique and proved the semistable College, for postgraduate studies. His research area was case of the modularity conjecture. number theory, the mathematical field that investigates The modularity lifting technique of Wiles concerns the properties of numbers. Under the guidance of his the Galois symmetries of the points of finite order in the advisor, John Coates, he studied elliptic curves, a type abelian group structure on an elliptic curve. Building upon of equation that was first studied in connection with Barry Mazur’s deformation theory for such Galois repre- measuring the lengths of planetary orbits. Together they sentations, Wiles identified a numerical criterion which made the first progress on one of the field’s fundamental ensures that modularity for points of order p can be lifted conjectures, the Birch and Swinnerton-Dyer conjecture, to modularity for points of order any power of p, where p proving it for certain special cases. Wiles was awarded is an odd prime. This lifted modularity is then sufficient his PhD in 1980 for the thesis “Reciprocity laws and the to prove that the elliptic curve is modular. The numerical conjecture of Birch and Swinnerton-Dyer”. criterion was confirmed in the semistable case by using an Between 1977 and 1980 he was an assistant professor important companion paper written jointly with Richard at Harvard University, where he started to study modular Taylor. Theorems of Robert Langlands and Jerrold Tunnell forms, a separate field from elliptic curves. There he began show that in many cases the Galois representation given a collaboration with Barry Mazur, which resulted in their by the points of order three is modular. By an ingenious 1984 proof of the main conjecture of Iwasawa theory, a switch from one prime to another, Wiles showed that in field within number theory. In 1982 he was made a profes- the remaining cases the Galois representation given by the sor at Princeton University. points of order five is modular. This completed his proof During the early years of his academic career he was of the modularity conjecture, and thus also of Fermat’s not actively trying to solve Fermat’s Last Theorem, nor Last Theorem. was anyone else, since the problem was generally regarded The new ideas introduced by Wiles were crucial to many as too difficult and possibly unsolvable. A turning point subsequent developments, including the proof in 2001 of came in 1986 when it was shown that the three-centuries- the general case of the modularity conjecture by Chris- old problem could be rephrased using the mathematics JUNE/JULY 2016 NOTICES OF THE AMS 609 of elliptic curves and modular forms. It was an amazing curves and Fermat’s Last Theorem”. As well as the atten- twist of fate that two subjects that Wiles had specialized tion of the global media, Wiles received many awards. in turned out to be exactly the areas that were needed They include the Rolf Schock Prize, the Ostrowski Prize, to tackle Fermat’s Last Theorem with modern tools. He the Wolf Prize, the Royal Medal of the Royal Society, the decided that he would return to the problem that so excited US National Academy of Science’s Award in Mathematics, him as a child. “The challenge proved irresistible,” he said. and the Shaw Prize. The International Mathematical Union Wiles made the unusual choice to work on Fermat presented him with a silver plaque, the only time they alone rather than collaborating with colleagues. Since the have ever done so. He was awarded the inaugural Clay problem was so famous, he was worried that news he was working on it would attract too much attention and he would lose focus. The only person he confided in was his wife, Nada, whom he married shortly after embarking on the proof. After seven years of intense and secret study, Wiles believed he had a proof. He decided to go public during a lecture series at a seminar in Cambridge, England. He did not announce it beforehand. The title of his talk, “Modular Forms, Elliptic Curves and Galois Representations”, gave nothing away, although rumor had spread around the mathematical community and two hundred people were packed into the lecture theater to hear him. When he wrote the theorem up as the conclusion to the talk, the room erupted in applause. Later that year, however, a referee checking the details Photographer: Heiko Junge/NTB scanpix The President of the Norwegian Academy of Science and of his proof found an error in it.